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sum_(k=0)^(99)2(3)^(k)~~ Choose 1 answer: (A) 5.15*10^(47) (B) 3.80*10^(30) (c) 2.37*10^(-30) (D) 7.76*10^(-48)

∑k=0992(3)k≈ \sum_{k=0}^{99} 2(3)^{k} \approx \newlineChoose 11 answer:\newline(A) 5.15⋅1047 5.15 \cdot 10^{47} \newline(B) 3.80⋅1030 3.80 \cdot 10^{30} \newline(C) 2.37⋅10−30 2.37 \cdot 10^{-30} \newline(D) 7.76⋅10−48 7.76 \cdot 10^{-48}

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Q. ∑k=0992(3)k≈ \sum_{k=0}^{99} 2(3)^{k} \approx \newlineChoose 11 answer:\newline(A) 5.15⋅1047 5.15 \cdot 10^{47} \newline(B) 3.80⋅1030 3.80 \cdot 10^{30} \newline(C) 2.37⋅10−30 2.37 \cdot 10^{-30} \newline(D) 7.76⋅10−48 7.76 \cdot 10^{-48}
  1. Identify series type: Identify the type of series.\newlineThe series ∑k=0992(3)k\sum_{k=0}^{99}2(3)^{k} is a geometric series because each term is obtained by multiplying the previous term by a common ratio, which in this case is 33.
  2. Use formula for sum: Use the formula for the sum of a finite geometric series.\newlineThe sum of a geometric series can be calculated using the formula Sn=a(1−rn)/(1−r)S_n = a(1 - r^n) / (1 - r), where aa is the first term, rr is the common ratio, and nn is the number of terms.
  3. Calculate first term: Calculate the first term aa of the series.\newlineThe first term when k=0k=0 is 2(3)0=2(1)=22(3)^0 = 2(1) = 2.
  4. Calculate number of terms: Calculate the number of terms nn in the series.\newlineSince the series starts at k=0k=0 and goes to k=99k=99, there are 100100 terms in total.
  5. Calculate sum using formula: Calculate the sum using the formula. S100=2(1−3100)1−3S_{100} = \frac{2(1 - 3^{100})}{1 - 3}
  6. Simplify expression: Simplify the expression.\newlineS100=2(1−3100)−2S_{100} = \frac{2(1 - 3^{100})}{-2}\newlineS100=−(1−3100)S_{100} = -(1 - 3^{100})
  7. Further simplify expression: Further simplify the expression. S100=3100−1S_{100} = 3^{100} - 1
  8. Evaluate expression: Evaluate the expression.\newlineSince 31003^{100} is a very large number, we can approximate it using scientific notation. However, we do not have the exact value of 31003^{100}, so we cannot directly calculate the answer. We need to estimate or use a calculator to find the value in scientific notation.
  9. Use calculator for scientific notation: Use a calculator or estimation to find 31003^{100} in scientific notation.\newlineAssuming we use a calculator, we find that 31003^{100} is approximately 5.1537752×10475.1537752 \times 10^{47}.
  10. Subtract 11 from calculated value: Subtract 11 from the calculated value to find the sum.\newlineS100=5.1537752×1047−1S_{100} = 5.1537752 \times 10^{47} - 1\newlineSince subtracting 11 from such a large number does not significantly change the value, we can say that S100S_{100} is approximately 5.15×10475.15 \times 10^{47}.

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